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Limits, Continuity, and Differentiability

Continuity A function is continuous on an interval if it is continuous at every point of the interval. Intuitively, a function is continuous if its graph can be drawn without ever needing to pick up the pencil. This means that the graph of y f (x) has no “holes”, no “jumps” and no vertical asymptotes at x = a. When answering free response questions on the AP exam, the formal definition of continuity is required. To earn all of the points on the free response question scoring rubric, all three of the following criteria need to be met, with work shown: A function is continuous at a point x = a if and only if:

1. f (a) exists 2. lim

xaf (x) exists

3. limxa

f (x) f (a) (i.e., the limit equals the function value)

Continuity and Differentiability Differentiability implies continuity (but not necessarily vice versa) If a function is differentiable at a point (at every point on an interval), then it is continuous at that point (on that interval). The converse is not always true: continuous functions may not be differentiable. It is possible for a function to be continuous at a specific value for a but not differentiable there. Example (graph with a sharp turn): Consider the function ( ) 3f x x :

The graph of ( ) 3f x x is continuous at 3x but the one-sided limits are not equal:

3

3 0( ) (3)lim 1

3 3x

xf x f

x x

and

3

3 0( ) (3)lim 1

3 3x

xf x f

x x

so f is not differentiable at 3x and the graph of f does not have a tangent line at the point (3, 0).

Student Study Session


Example (graph with a vertical tangent line) Consider the function 1

3( ) 2f x x :

The function 1

3( ) 2f x x is continuous at 0x but because the limit

1

3

20 0 03

( ) (0) 2 0 2lim lim lim

03

x x x

f x f x

x xx

is infinite, it can be concluded that the tangent line is vertical at 0x , therefore, f is not differentiable at 0x . Quick Check for Understanding: 1. Sketch a function with the property that f (a) exists but lim

xaf (x) does not exist.

2. Sketch a function with the property that lim

xaf (x) exists but f (a) does not exist.

3. Sketch a function with the property that f (a) exists and lim

xaf (x) exists but lim

xaf (x) f (a).

Limits, Continuity, and Differentiability Student Study Session


Students should be able to:

Determine limits from a graph Know the relationship between limits and asymptotes (i.e., limits that become infinite at a

finite value or finite limits at infinity) Compute limits algebraically Discuss continuity algebraically and graphically and know its relation to limit. Discuss differentiability algebraically and graphically and know its relation to limits and

continuity Recognize the limit definition of derivative and be able to identify the function involved

and the point at which the derivative is evaluated. For example, since

, recognize that is simply the

derivative of cos(x) at . L’Hôpital’s Rule (BC only) Limits associated with logistic equations (BC only)

f (a) limh0

f (a h) f (a)

hlimh0

cos( h) cos( )

hx



Multiple Choice 1. (calculator not allowed) (1985AB5)

2

2

4lim

10000n

n

n n is

(A) 0

(B) 1

2500

(C) 1 (D) 4 (E) nonexistent 2. (calculator not allowed)

(2 1)(3 )

lim( 1)( 3)x

x x

x x

is

(A) 3 (B) 2 (C) 2 (D) 3 (E) nonexistent 3. (calculator not allowed)

What is

8 8

0

1 18 8

2 2limh

h

h

?

(A) 0

(B)

1

2 (C) 1 (D) The limit does not exist. (E) It cannot be determined from the information given. 4. (calculator not allowed)

4

sin4lim

4x

x

x

is

(A) 0

(B) 1

2

(C) 4

(D) 1 (E) nonexistent



5. (calculator not allowed)

The 0

tan 3( ) tan 3limh

x h x

h

is

(A) 0 (B) 23sec (3 )x

(C) 2sec (3 )x (D) 3cot(3 )x (E) nonexistent 6. (calculator not allowed)

If 2( ) 2 1f x x , then 20

( ) (0)limx

f x f

x

is

(A) 0 (B) 1 (C) 2 (D) 4 (E) nonexistent


If ( ) cosf x x and ( ) 1g x for all x, and if (0) (0) 0f g , then 0

( )lim

( )x

f x

g x is

(A) 2

(B) 1

(C) 0

(D) –1

(E) nonexistent




20

1 coslim

2sin

is

(A) 0

(B) 1

8

(C) 1

4

(D) 1

(E) nonexistent 9. (calculator not allowed) If lim ( )

x af x L

where L is a real number, which of the following must be true?

(A) '( )f a exists. (B) ( )f x is continuous at x a . (C) ( )f x is defined at x a . (D) ( )f a L (E) None of the above 10. (calculator not allowed) For 0x , the horizontal line 2y is an asymptote for the graph of the function f . Which

of the following statements must be true? (A) (0) 2f

(B) ( ) 2f x for all 0x (C) (2)f is undefined.

(D) 2

lim ( )x

f x

(E) lim ( ) 2x

f x




If the graph of ax b

yx c

has a horizontal asymptote at 2y and a vertical asymptote at

3x , then a c (A) 5 (B) 1 (C) 0 (D) 1 (E) 5 12. (calculator not allowed)

At 3x , the function given by 2 , 3

( )6 9, 3

x xf x

x x

is

(A) undefined. (B) continuous but not differentiable. (C) differentiable but not continuous. (D) neither continuous nor differentiable. (E) both continuous and differentiable. 13. (calculator not allowed)

f (x) x 2 if x 34x 7 if x 3

Let f be the function given above. Which of the following statements are true about f ? I.

3lim ( )x

f x

exists.

II. f is continuous at 3x .

III. f is differentiable at 3x . (A) None (B) I only (C) II only (D) I and II only (E) I, II and III




2 4, 2

( ) 21, 2

xx

f x xx

Let f be the function defined above. Which of the following statements about f are true? I. f has a limit at 2x . II. f is continuous at 2x . III. f is differentiable at 2x . (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III 15. (calculator not allowed) Let f be defined by the following.

2

sin , 0

, 0 1( )

2 , 1 2

3, 2

x x

x xf x

x x

x x

For what values of x is f NOT continuous? (A) 0 only (B) 1 only (C) 2 only (D) 0 and 2 only (E) 0, 1, and 2 16. (calculator not allowed) If for all x, then the value of the derivative at x = 3 is

(A) 1 (B) 0 (C) 1 (D) 2 (E) nonexistent

f (x) 2 x 3 f (x)




If the function f is continuous for all real numbers and if 2 4

( )2

xf x

x

when 2x ,

then ( 2)f (A) 4 (B) 2 (C) 1 (D) 0 (E) 2 18. (calculator not allowed) The graph of the function f is shown in the figure above. Which of the following statements

about f is true? (A) lim ( ) lim ( )

x a x bf x f x

(B) lim ( ) 2x a

f x

(C) lim ( ) 2x b

f x

(D) lim ( ) 1x b

f x

(E) lim ( )x a

f x

does not exist.




The graph of a function f is shown above. At which value of x is f continuous, but not

differentiable? (A) a (B) b (C) c (D) d (E) e

20. (calculator allowed)

For which of the following does

4lim ( )x

f x

exist?

(A) I only (B) II only (C) III only (D) I and II only (E) I and III only



21. (calculator allowed) The figure above shows the graph of a function f with domain 0 4x . Which of the

following statements are true? I.

2lim ( )x

f x

exists

II. 2

lim ( )x

f x

exists

III. 2

lim ( )x

f x

exists

(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III 22. (calculator allowed) The graph of the function f is shown in the figure above. The value of

1lim sin isx

f x

(A) 0.909 (B) 0.841 (C) 0.141 (D) –0.416 (E) nonexistent



Free Response 23. (calculator allowed)

A continuous function f is defined on the closed interval 4 6x . The graph of f

consists of a line segment and a curve that is tangent to the x-axis at 3x , as shown in the figure above. On the interval 0 6x , the function f is twice differentiable, with

"( ) 0f x . (a) Is f differentiable at 0x ? Use the definition of the derivative with one-sided limits to

justify your answer.



24. (calculator not allowed) Let f be the function defined by

1 for 0 3

( )5 for 3 5.

x xf x

x x

(a) Is f continuous at 3x ? Explain why or why not. (c) Suppose the function g is defined by

1 for 0 3

( )2 for 3 5,

k x xg x

mx x

where k and m are constants. If g is differentiable at x = 3, what are the values

of k and m?




Let f be a function defined by 4

1 2sin , 0( )

, 0.x

x xf x

e x

(a) Show that f is continuous at 0.x (b) For 0,x express ( )f x as a piecewise-defined function. Find the value of x for which

( ) 3.f x



26. (calculator not allowed) 2012 AB 4

The function f is defined by for

(c) Let g be the function defined by

Is g continuous at Use the definition of continuity to explain your answer.

2

2012 AB 4 Rubric

The function f is defined by for

(c) Let g be the function defined by

Is g continuous at Use the definition of continuity to explain your answer.

(c)

Therefore,

So,

1: considers one-sided limits 1: answer with explanation

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Limits, Continuity, and Differentiability - Weeblygvmath.weebly.com/uploads/5/6/5/3/56539921/ap_calc.limits_contin… · Limits, Continuity, and Differentiability Continuity A function